Question

I need help. I dont get how i find teh domain and range. do I graph it first?

Answer 1

Answer:

Domain: $-1 \le x < \infty$ or in interval notation $[-1, \infty)$

Range: $-\infty < x < 0$ or, in interval notation $(-\infty, 0)$

And yes, you can graph it but not necessary for a  problem like this. I have provided the graph

Step-by-step explanation:

This is known as a piecewise function. Two different expressions for two different intervals

The domain of a function f(x) is the range of input values for which f(x) is real and defined

For example, the function $f(x) = \frac{1}{x+1}$ is the set of all x values except for x = -1. Because at x = -1, the denominator will be 0 and division by 0 does not result in a real value for f(x). (It is actually defined as $\infty$)

For $f(x) = x^2 - 1$ , the domain is $-\infty < x < \infty$ since either negative x or positive x will result in a real value for the function but not at $-\infty$ or $\infty$

But the above function is only defined for $-1 \le x \le 1$. So if we consider this part of the function the domain is $-1 \le x \le 1$

expressed in interval notation: [-1, 1]

The range is max-min.

At x = 0, min = -1, at x = ∞, max = ∞

So range of this part of the piecewise function is [-1, ∞)

For the second interval f(x) = $-x + 1$ at $x > 1$ . Since x can range from > 1 to $\infty$,  the function is valid only for 1 <x < $\infty$

So the domain of this part of the function is

$1 \le x < \infty$   expressed in interval notation as $(1, \infty)$

For range,

At x = 1 , the function value is -1 + 1 = 0 (max)

At x = $\infty$, the function value is $-\infty$ (min)

So range of this part of the function is (-∞, 0)

Taking both into consideration, we choose the most appropriate lowest and highest levels. This is called the union of the two domains.

Hence the domain for the entire piecewise function is

[-1, ∞)

Range of the entire piecewise function is

(-∞, 0)

(I had to do a lot of typing here. Not sure if I made typos. Please check)